Fuzzy modeling consequence part

Inference Engine Inference Engine maps input type-2 fuzzy sets into output type-2 fuzzy sets by applying the consequent part where this process of mapping from the antecedent part into the consequent part is interpreted as a type-2 fuzzy implication which needs computations of union and intersection of type-2 fuzzy sets. The inference engine in Mamdani system maps the input fuzzy sets into the output fuzzy sets then the defuzzifier converts them to crisp outputs. The rules in Mamdani model have fuzzy sets in both the antecedent part and the consequent part. For example, the /th rule in a Mamdani rule base can be described as follows ... 

Takagi and Sugeno (1985) proposed models where the consequent part of the rule is described by a linear regression model. These models are easier to identify because each rale describes a fuzzy region in which the output depends on the inputs in a linear maimer. An example of such a model is shown in Eqn. (28.3) ... 

In view of the linear form of the consequence part use in fuzzy models, an obvious choice for fuzzy clustering is the fuzzy C-varieties or Gustafson-Kessel algorithm, in which linear or planar clusters are allowed as prototypes to be sought. 

The algorithms summarized in the previous two sections can easily be applied to identify the consequence part of a fuzzy Takagi-Sugeno model. 

The fuzzy model can now be completed by calculating the consequence part parameters using the least squares approach. This yields the following local linear models ... 

Although it is possible to optimize the fuzzy models for // and r sequentially, it is interesting to see how the large-scale algorithm deals with optimization of a large set of parameters in a hybrid model. Therefore, all the parameters of the two fuzzy models will be optimized simultaneously. The result is that a set of 66 parameters will be optimized 48 premise part parameters and 18 consequence part parameters. The premise part parameters are constrained the bounds are set at the initial values 10%. No constraints were placed on the consequence part parameters. The results of the optimization are shown in Fig. 30.11. (for clarity, only some of the measurements are shown). The anomalous behavior has been removed and model offset has been reduced to acceptable levels. 

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